rastafarral6

2021-11-13

Hi i know this is a really really simple question but it has me confused.

I want to calculate the cross product of two vectors

$\overrightarrow{a}\times \overrightarrow{r}$

The vectors are given by

$\overrightarrow{a}=a\overrightarrow{z},\text{}\overrightarrow{r}=x\overrightarrow{x}+y\overrightarrow{y}+z\overrightarrow{z}$

The vector$\overrightarrow{r}$ is the radius vector in cartesian coordinares.

I want to calculate the cross product in cylindrical coordinates, so I need to write$\overrightarrow{r}$ in this coordinate system.

The cross product in cartesian coordinates is

$\overrightarrow{a}\times \overrightarrow{r}=-ay\overrightarrow{x}+ax\overrightarrow{y}$ ,

however how can we do this in cylindrical coordinates?

I want to calculate the cross product of two vectors

The vectors are given by

The vector

I want to calculate the cross product in cylindrical coordinates, so I need to write

The cross product in cartesian coordinates is

however how can we do this in cylindrical coordinates?

Donald Proulx

Beginner2021-11-14Added 18 answers

The radius vector $\overrightarrow{r}$ in cylindrical coordinates is $\overrightarrow{r}=p\overrightarrow{p}+z\overrightarrow{z}$ . Calculating the cross-product is then just a matter of vector algebra:

$\overrightarrow{a}\times \overrightarrow{r}=a\overrightarrow{z}\times (p\overrightarrow{p}+z\overrightarrow{z})$

$=a(p(\overrightarrow{z}\times \overrightarrow{p})+z(\overrightarrow{z}\times \overrightarrow{z}))$

$=ap(\overrightarrow{z}\times \overrightarrow{p})$

$=ap\overrightarrow{\varphi}$ ,

where in the last line we've used the orthonormality of the triad$\{\overrightarrow{p},\overrightarrow{\varphi},\overrightarrow{z}\}$

where in the last line we've used the orthonormality of the triad